Logarithm and Antilogarithm Test 21

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Logarithm and Antilogarithm Test 21

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Weekly Quiz Competition

Question 1
1 / -0

If the approximate value of $$ \log _{ 10 }{ (4.04) }$$is $$0. \ abcdef,$$ it is given that $$\log _{ 10 }{ (4) }=0.6021$$ &$$ \log _{ 10 }{ (e) }=0.4343,$$ then the value of abcd must be

A
$$6064$$

B
$$6063$$

C
$$6065$$

D
N.O.T

Solution
Question 2
1 / -0

If $$\log_{10}e=0.4343$$, then $$\log_{10}1016$$ is

A
$$2.99$$

B
$$3$$

C
$$3.006949$$

D
$$3.02$$

Solution

$$\log_{10}1016\Rightarrow \dfrac{\log 1016}{\log 10}$$
$$\Rightarrow \dfrac{3.006949}{1}$$
$$\rightarrow$$ Option $$C$$ is correct
Question 3
1 / -0

The number of zeroes after decimal and before first significant digit in $$(50)^{-100}$$ is equal to : (take $$log_{10}$$ 5=0.699)

A
168

B
169

C
170

D
171

Solution

So, the correct answer is '169'.
Question 4
1 / -0

$$2^{74}-2^{73}-2^{72}$$ is same as

A
$$2^{72}$$

B
$$2^{71}$$

C
$$2^{70}$$

D
$$none\ of\ these$$

Solution

$$2^{74}-2^{73}-2^{72}=2^{72}(2^2-2-1)=2^{72}\times 1=2^{72}$$
Hence,correct option is $$A.$$
Question 5
1 / -0

Find the value of $$\log_{10}{\left(0.\bar{9}\right)}$$

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$2$$

Solution

To find value of $$\log_{10}{0.\bar9}$$

Let $$x=0.\bar{9}=0.999999...$$

$$\Rightarrow 10x=9.99999....$$

$$\Rightarrow 10x-x=9$$

$$\Rightarrow 9x=9$$

$$\Rightarrow x=\dfrac{9}{9}=1$$

$$\therefore x=1$$

Let $$y=\log_{10}{1}$$

$$\Rightarrow 1={10}^{y}$$

$$\Rightarrow {10}^{y}={10}^{0}$$ (since $${10}^{0}=1$$)

Since bases are same we can equate the powers

$$\therefore y=0$$

Hence, $$\log_{10}{\left(0.\bar{9}\right)}=0$$
Question 6
1 / -0

If $$\log 4=1.3868$$, then the approximate value of $$\log\, (4.01)$$

A
$$1.3968$$

B
$$1.3898$$

C
$$1.3893$$

D
$$1.9338$$

Solution

Let $$y=f(x)=\log x$$

Let $$x=4$$

$$X+\triangle x=4.01$$

$$\triangle x=0.01$$

For $$x=4$$

$$Y=\log4=1.3868$$

$$y=\log x$$

$$\dfrac{dy}{dx}=\dfrac{1}{x}=\dfrac{1}{4}$$

$$\triangle y=dy$$

$$=\dfrac{dy}{dx}.dx$$

$$=\dfrac{1}{4}\times 0.01$$

$$\triangle y=0.0025$$

$$\log(4.01)=y+\triangle y$$$$=1.3893$$
Question 7
1 / -0

If $$x=500,y=100$$ and $$z=5050$$, then the value of $$(\log _{ xyz }{ { x }^{ z } } )(1+\log _{ x }{ yz } )$$ is equal to.

A
500

B
100

C
5050

D
10

Solution

Given,

$$\left(\log _{xyz}\left(x^z\right)\right)\left(1+\log _x\left(yz\right)\right)$$

$$\left(\log _{xyz}\left(x^z\right)\right)\left(1+\log _x\left(yz\right)\right)$$

$$=z\log _{xzy}\left(x\right)\left(\log _x\left(zy\right)+1\right)$$

from given w have,

$$=5050\log _{(500 \times 5050 \times 100)}\left(500\right)\left(\log _{500}\left((5050 \times 100)\right)+1\right)$$

$$=\dfrac{5050\log_e \left(505000\right)}{\log_e \left(252500000\right)}+5050\log _{252500000}\left(500\right)$$

$$=5050$$
Question 8
1 / -0

The greatest value of $$(4\log_{10}{x}-\log_{x}{(0.0001)})$$ for $$0 < x < 1$$ is

A
$$4$$

B
$$-4$$

C
$$8$$

D
$$-8$$

Solution

$$(4log_{10}x-log_x(0.0001)$$
$$(4log_{10}x-log_{x}\dfrac{1}{10^4}$$
$$(4log_{10}x-(log_x1-log_x{10^4})$$
$$(4log_{10}x-0+log_x{10^4})$$
$$(4log_{10}x+log_x{10^4})$$
$$4(log_{10}x+log_x10)$$
$$4(log_{10}x+\dfrac{1}{log_{10}x})$$
$$AM\geq GM$$
$$\dfrac{x+y}{2} \geq \sqrt{xy}$$
$$log_{10}x+\dfrac{1}{log_{10}x} \geq 2$$
$$4(log_{10}x+\dfrac{1}{log_{10}x}) \geq 8$$
Question 9
1 / -0

The value of $$ 3 ^{log_4 5} -5 ^{log_4 3}$$

A
$$0$$

B
$$1$$

C
$$2$$

D
None of these

Solution
Question 10
1 / -0

The number of $$\log_2 7 $$ is

A
an integer

B
a rational number

C
an irrational number

D
a prime number

Solution

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Re-Attempt Weekly Quiz Competition

Logarithm and Antilogarithm Test 21

Result

Logarithm and Antilogarithm Test 21

Score

-

Rank

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SHARING IS CARING

Question 1

$$6064$$

$$6063$$

$$6065$$

N.O.T

Solution

Question 2

$$2.99$$

$$3$$

$$3.006949$$

$$3.02$$

Solution

Question 3

168

169

170

171

Solution

Question 4

$$2^{72}$$

$$2^{71}$$

$$2^{70}$$

$$none\ of\ these$$

Solution

Question 5

$$0$$

$$1$$

$$-1$$

$$2$$

Solution

Question 6

$$1.3968$$

$$1.3898$$

$$1.3893$$

$$1.9338$$

Solution

Question 7

500

100

5050

10

Solution

Question 8

$$4$$

$$-4$$

$$8$$

$$-8$$

Solution

Question 9

$$0$$

$$1$$

$$2$$

None of these

Solution

Question 10

an integer

a rational number

an irrational number

a prime number

Solution

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